Exercise 12, § 1

(a) Let $A$ be an integrally closed domain, $f(X) = \sum_i a_i X^i$ and $g(X) = \sum_j b_j X^j$ be two polynomials in $A[X]$; show that, if $c \in A$ is such that all the coefficientsoff $(X)g(X)$ belong to $Ac$, then all the products $a_i b_j$ belong to $Ac$ (reduce it to the case where $A$ is a valuation ring).

(b) If $A$ is the ring defined in Exercise 1, give an example of two polynomials $f, g$ of degree 1 in $A[X]$ and an element $c \in A$ for which the conclusion of (a) does not hold (take $c = T^4 + T^5$).

(c) Under the hypotheses of (a), let $a, b, c$ be the ideals of $A$ generated respectively by the coefficientsoff, $g$ and $fg$: show that $\operatorname{div}(c) = \operatorname{div}(a) + \operatorname{div}(b)$. Give an example where $\tilde{c} \neq \tilde{a} \tilde{b}$. Show that, if $c$ is principal, $c = ab$ and $a$ and $b$ are invertible; if further $A$ is a local ring, $a$ and $b$ are principal.